Understanding the properties of numbers, particularly the number of factors and the sum of factors, is a cornerstone of quantitative aptitude for competitive exams like the CAT. This module will equip you with the foundational knowledge and techniques to efficiently solve problems related to these concepts.

The first and most crucial step in determining the number of factors and the sum of factors of any integer is its prime factorization. Every integer greater than 1 can be uniquely expressed as a product of prime numbers. For example, the prime factorization of 12 is $2^2 \times 3^1$.

Once a number 'N' is expressed in its prime factored form as $N = p_1^{a_1} \times p_2^{a_2} \times \dots \times p_k^{a_k}$, where $p_1, p_2, \dots, p_k$ are distinct prime numbers and $a_1, a_2, \dots, a_k$ are their respective positive integer exponents, the total number of factors (or divisors) of N is given by the product of one more than each exponent.

Number of Factors = $(a_1 + 1)(a_2 + 1)\dots(a_k + 1)$

The sum of factors of a number N, when expressed as $N = p_1^{a_1} \times p_2^{a_2} \times \dots \times p_k^{a_k}$, is calculated by summing all possible combinations of its prime factors raised to powers from 0 up to their respective exponents. This can be expressed as:

Sum of Factors = $(p_1^0 + p_1^1 + \dots + p_1^{a_1}) \times (p_2^0 + p_2^1 + \dots + p_2^{a_2}) \times \dots \times (p_k^0 + p_k^1 + \dots + p_k^{a_k})$

Each term in parentheses is a geometric progression, which can be simplified. For example, $(p^0 + p^1 + \dots + p^a) = \frac{p^{a+1}-1}{p-1}$.

A number is prime if it has exactly two factors: 1 and itself. A perfect number is a positive integer that is equal to the sum of its proper positive divisors (divisors excluding the number itself). For example, 6 is a perfect number because its proper divisors are 1, 2, and 3, and $1+2+3=6$.